Simplify Square Root of 76545

Here we will show you two methods that you can use to simplify the square root of 76545. In other words, we will show you how to find the square root of 76545 in its simplest radical form using two different methods.

To be more specific, we have created an illustration below showing what we want to calculate. Our goal is to make "A" outside the radical (√) as large as possible, and "B" inside the radical (√) as small as possible.

√76545 = A√B

Greatest Perfect Square Factor Method
The Greatest Perfect Square Factor Method uses the greatest perfect square factor of 76545 to simplify the square root of 76545. This is how to calculate A and B using this method:

A = Calculate the square root of the greatest perfect square from the list of all factors of 76545. The factors of 76545 are 1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 81, 105, 135, 189, 243, 315, 405, 567, 729, 945, 1215, 1701, 2187, 2835, 3645, 5103, 8505, 10935, 15309, 25515, and 76545. Furthermore, the greatest perfect square on this list is 729 and the square root of 729 is 27. Therefore, A equals 27.

B = Calculate 76545 divided by the greatest perfect square from the list of all factors of 76545. We determined above that the greatest perfect square from the list of all factors of 76545 is 729. Furthermore, 76545 divided by 729 is 105, therefore B equals 105.

Now we have A and B and can get our answer to 76545 in its simplest radical form as follows:

√76545 = A√B

√76545 = 27√105

Double Prime Factor Method
The Double Prime Factor Method uses the prime factors of 76545 to simplify the square root of 76545 to its simplest form possible. This is how to calculate A and B using this method:

A = Multiply all the double prime factors (pairs) of 76545 and then take the square root of that product. The prime factors that multiply together to make 76545 are 3 x 3 x 3 x 3 x 3 x 3 x 3 x 5 x 7. When we strip out the pairs only, we get 3 x 3 x 3 x 3 x 3 x 3 = 729 and the square root of 729 is 27. Therefore, A equals 27.

B = Divide 76545 by the number (A) squared. 27 squared is 729 and 76545 divided by 729 is 105. Therefore, B equals 105.

Once again we have A and B and can get our answer to 76545 in its simplest radical form as follows:

√76545 = A√B

√76545 = 27√105

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